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midterm2-practice

# midterm2-practice - MATH 131B 2ND PRACTICE MIDTERM Problem...

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Unformatted text preview: MATH 131B 2ND PRACTICE MIDTERM Problem 1. State the book’s deﬁnition of: (a) A complete metric space (b) and (c) Convergence of a series of real numbers (d) Normed vector space; Banach space Problem 2. Let be a metric space with a metric . Let and be two Cauchy sequences in . Show that exists. Note: we do not assume that is complete. is closed if Problem 3. Let be the usual Eucledian metric on . We say that a subset whenever and , then . Show that a subset is complete with respect to if and only if it is closed. Problem 4. Let be given by  ¡ ¢¢¢ £¡      3   )  &  4210('%\$" #!  5 76  5 76 5 © §¥ ¨¦¤¢ £¡ £¡ ¤¢     C¦ 9 5 BB 9 @ A8   %8 9   G5 QPIH4FD @ G 5E `3 qp bhBg1"eBc2bB\$W7R1TSRD i` Y X f )  d` Y X  a` Y X V& U 3 ) & Show that there is a unique point with the property that . Problem 5. State and prove that Banach contraction principle. Problem 6. Let and be norms on the space of continuous functions on the , given by: interval G 3 r ) r & q4¨s8Sxw141sSvD U 3r ) r & 5 utq41sS& 9 3r)r   d ) Y "h y D "Qy #y D Qy d)Y "   3 & pSRD ©§ ¨¥  ` T U   '¨r  "   #y D Qy   3 & p'RD U "Qy y D r f f U ed  f  ¡ h¢¦¤¢ £¡ f 1 if and only if  Show that the two norms are not equivalent. Problem 7. Let and . Show that and moreover that if this is the case, then . Problem 8. State and prove the comparison test. converges, @ i f U f  f © §¥ g0¨¦¤¢ £¡ U ed ...
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